Computing inbreeding coefficients in large populations

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Computing inbreeding coefficients in large populations

The Meuwissen, Z Luo

To cite this version:

Original

article

Computing

inbreeding

coefficients

in

_{large populations}

THE Meuwissen

Z Luo

Institute

_of

Animal

_Physiology

and Genetics

_Research,

Edinburgh

Research

Station,

Roslin EH25

9PS,

UK

(Received

21 October 1991;

accepted

15

_April

₁₉₉₂₎

Summary - An

_algorithm

for

computing inbreeding

coefficients in

_{large populations}

is

presented.

It is

_especially

useful in

_{large populations}

because of the small size of the

memory

required,

which is linear with

_population

size, and its

_speed,

if the number of

generations

involved is not too

_large,

ie not

_larger

than about 12. The method is

_compared

with 2 other methods for

_{computational speed}

and memory requirement. The

_presented

algorithm

is suited for situations where the

_inbreeding

coefficients for a few new animals are to be

computed given

that their ancestor’s

inbreeding

coefficients were calculated

previously.

inbreeding coefficient

_/

_algorithm

Résumé - Le calcul des coefficients de consanguinité dans de grandes populations.

On

_présente

un

_algorithme

de calcul des

_coefficients

de

_{consanguinité}

pour de

grandes

populations.

Il est

particulièrement adapté

à ces

populations

à cause du

faible

volume de

mémoire d’ordinateur _requis

_(il

_dépend

linéairement de la taille de la

_population)

et de la rapidité de calcul

_quand

le nombre de

_{générations impliquées}

n’est pas trop élevé

(c’est-à-dire

inférieur

à environ

_12).

La méthode est

_comparée

à 2 autres méthodes du point de

vue de la vitesse de calcul et de la mémoire

_requise.

Le

_{présent algorithme}

est

_également

adapté

aux situations de mise à

_jour

où les

_coefficients

de

_{consanguinité correspondant}

à un

_faible

nombre d’animaux nouveaux doit être calculé en tirant _parti des

coefficients

calculés pour les anciens animaux.

coefficient de _{consanguinité}

_/

_algorithme

*

On leave from the Schoonoord Research Institute for Animal

_Production,

PO Box

_501,

INTRODUCTION

Many

countries are

_implementing

or have

_implemented

evaluation methods for

na-tional herds based upon animal models. If these account for

_inbreeding,

calculation of

_inbreeding

coefficients in

_{large populations}

is

_required.

Henderson

₍₁₉₇₆₎

and

Quaas

(1976)

implicitly

present

methods for the calculation of

_inbreeding

coeffi-cients,

when

they

propose

algorithms

for the calculation of the inverse of the

ad-ditive

_relationship

matrix. Henderson’s method

_requires

_storage

of a

large

matrix.

Quaas

avoids this: memory

requirement

is linear with N and

computation

time is

proportional

to

_N

₂

_,

where N is the size of the data set. In

Quaas’

method

com-puter

time becomes a

limiting

factor with

_increasing

N. With this

algorithm,

no

use is made of known

_inbreeding

coefficients,

and hence all calculations have to be

repeated

whenever coefficients are

_required

for a new batch of animals. Golden et al

₍₁₉₉₁₎

_present

an

algorithm

based on

Quaas’

method

_using

_sparse

_programming

techniques.

Tier

₍₁₉₉₀₎

presents

a fast method for the calculation of

_inbreeding

coefficients.

Using

sparse

programming

techniques,

his

_algorithm

first determines which ele-ments in the additive

_{genetic relationship}

matrix A are

required

and then calculates

them. The _memory

_required

is a small

_proportion

of

NZ:

about

0.9%.

For

_large

pop-ulations,

the

_physical

memory of the

computer

is still

_likely

to be too small and use

of disk _{memory is}

_{required, slowing}

this

_{algorithm considerably.}

Known

_inbreeding

coefficients are not recalculated.

The aim of this _{paper is} to

_present

a method for the calculation of

_inbreeding

coefficients in

_{large populations,}

which is

_{fast, requires}

_memory

_proportional

to N and does not

_recompute

known

_inbreeding

coefficients. The method

_presented

is

compared

to Golden et al’s

₍₁₉₉₁₎

_{implementation}

of

_Quaas’

₍₁₉₇₆₎

method and Tier’s

₍₁₉₉₀₎

method for

_speed

and _memory

_required.

METHOD

The method is based on the

_{decomposition}

of the additive

_{genetic relationship}

matrix

_A,

as described

_by

Henderson

_(1976):

A =

LDL’,

where L is _a lower

triangular

matrix

_containing

the fraction of the genes that animals derive from

their

_ancestors,

and D is a

_diagonal

matrix

_containing

the within

_family

additive

genetic

variances of animals. The animals are ordered such that

_parents

precede

offspring.

From the

_{decomposition,}

it follows that

_{(Quaas, 197G):}

where

_A

_ii

is the ith

_diagonal

element of

_A,

which

_equals

the

_inbreeding

coefficient of animal i

_plus

1.

_Quaas

₍₁₉₇₆₎

_computes

the elements of L

_quickly

and one column at a time

_by

tlie

_following

recursive

_algorithm:

For

_j=

1 to 1V

_(all

columns of

_L)

For

i= j

+

1 to N

_(all

elements below the

_diagonal

_element)

Lij =

(L!

+

_L

_dd

_)/2

when both

parents

si and

d

i

of i are

_known,

or

=

L

kd

/2

when

_only

one

_parent

k

i

of i is

known,

or

= 0 when both

_parents

are

_unknown,

for i <

_j.

The elements of D are calculated as:

D!!

= 1 when both

_parents

are

_unknown,

or

= 0.75 -

Fkj /4

when

_only

one

_parent

k!

of j

is

known,

or

= 0.5 -

(Fs!

+

_F

_dJ/

₄

when both

_parents

_Sj and

_d

_j

of j

are

known,

where

F

j

denotes the

_inbreeding

coefficient of animal

_j.

After

_computing

the elements of the

_jtli

column of

_{L, they}

are

squared

and

multiplied by

_D!!.

The

resulting

vector is added to a

_working

vector. When this

_procedure

is followed for _every column of

L,

the

working

vector contains the

_A

_jj

_-values.

This

algorithm requires

N(N+ 1)/2

operations.

Golden et al

₍₁₉₉₁₎

made a list of the

_offspring

of all the animals i.

Because element

_L

_ij

is non-zero

_only

if i is a descendant

_{of j,}

the

_operations

within

each column are

performed only

for the descendants

of j.

Because the elements of

the columns of L are not stored

_they

have to be recalculated when a new batch of

animals becomes available.

In the

present

algorithm,

the elements of L are

computed

row

by

row, which

overcomes the

problem

of

recalculating

elements of L when a new batch of animals

has to be evaluated. A row i of L

_gives

the fraction of the _genes that animal i derives from its ancestors.

_Hence,

_L

_isi

=

Lid!

=

0.5,

where _si and

d

i

are the sire and dam of

i,

respectively.

Each row of L can be built

_{by proceeding}

up the

pedigree adding

half the &dquo;contribution&dquo; of the current animal to each of its

_parents.

The fraction of the _genes derived from an ancestor is:

where

_P

_j

is the set of identification numbers of the progeny

of j

and

ANC

i

is the set of identification numbers of the ancestors of

_i,

_including

animal i itself. The latter

_identity

in

_[2]

is because

_Li!

=

0,

if k is not an ancestor of i or not

_equal

to i. This leads to the

_{following algorithm}

for

_{computing L,}

row

_by

row. As each row is

_determined,

the contributions to the elements of

A

ii

are accumulated. Row

i of L and

A;z,

are set to 0 before

_starting.

The

_{algorithm keeps}

track of a list of

ancestors

_ANC

_I

_,

whose contribution to

_A

_ii

has

_yet

to be included. If the sire or dam is unknown si = 0 _or

d

i

=

_0,

respectively.

delete j

from

_ANC,

end wliile

The

_youngest

_{animal j}

in

_ANC

_i

is

_evaluated,

because all of its _progeny in the

pedigree

of animal i must have been

evaluated, hence,

its

_L

_ij

value is known. The kernel of a Fortran _program of the

_algorithm

is

given in

the

_Appendix.

The list of ancestors is

_{represented by}

a link list. In the

_computer

_program time is saved

_by:

_i)

checking

whether one of the

_parents

is

unknown, giving

an

inbreeding

coefficient of zero

(otherwise,

the

_algorithm

would trace the

_pedigree

of the other

_{parent) ;}

and

_ii)

if the

_pedigree

is sorted such that full sibs have successive identification

numbers,

ie

are evaluated

_{successively,}

_only

the first full sib is evaluated and all full sibs obtain

(and have)

the same

_inbreeding

coefficient as the evaluated full sib.

In order to _compare the

_algorithms,

Golden et al’s

₍₁₉₉₁₎

and Tier’s

₍₁₉₉₀₎

algorithms

were

_programmed

in Fortran. Golden et al

_presented

their

_algorithm

in

C,

which

_they

considered faster.

However,

C was not available to the authors. The

sorting

of the list of descendants in the Golden et al

_algorithm

was

performed by

an IMSL routine here

(I1VISL, 1984).

SIMULATION

In order to _compare the

_algorithms,

the simulation program of Meuwissen and Van

der Werf

₍₁₉₉₁₎

was used to

_generate

a

_pedigree

data set. The _program simulated

an _open

_dairy

cattle nucleus scheme for

10,

20 or 40 _years, with

population

sizes of

9 267,

15 582 and 28171

_{animals, respectively}

_(see

table

_I).

Selection was for animal

model BLUP

_breeding

values. Nucleus dams and commercial dams

_produced

8 and

1

_{offspring, respectively.}

The number of nucleus and commercial sires selected was

4 and

10,

respectively. Also,

a

_breeding

_program was simulated for 20 _years with

selection of

_only

one commercial

_sire,

_representing

a situation with

_large

half sib families.

Inbreeding

coefficients were calculated on a micro-VAX 3800 in batch mode. The maximum

_physicsl

memory allocated to the batch _queue was about 20 Mb. Because

this exceeded the _memory

_{required by}

_any of the

_algorithms,

no disk _memory was

used. If disk _memory were

used,

the

_{computational speed}

would

_depend

on the size

The number of

_generations

in the data-set after

10,

20 and 40 _years were

counted and a

_weighted

_{average per} individual was

calculated,

where

_weighting

was

according

to the contribution of the

_path.

For

_instance,

if

_only

the sire of the

sire and the sire of an animal were

known,

the _average number of

_generations

was

2.1/4

+

_{1 ! 1/4}

+

_{0 -1/2}

= 0.75. The animal with the maximum average number of

generations

and the _average number of

_generations

of the animals born in the last

year evaluated were estimated.

RESULTS

The results are

presented

in table II.

Although

the

algorithm

of Golden et al

(1991)

and that

_presented

here

_require

the same number of additions of

L !D!!

terms, the latter consumed less

_computer

time. This is because the

_algorithm

of Golden et al:

_i)

makes a list of

_offspring

of each

animal,

before

_{starting evaluations;}

ii)

traces descendants instead of

ancestors,

which is more difficult because the

number of progeny of each animal is

unlimited,

whereas the number of

parents

is limited

to 2;

_iii)

sorts the list of descendants for _every

_animal,

which is

time-consuming ;

and

_iv)

_tracing

of

_{descendants, sorting them,}

and addition of

_{L? -}

_D

_jj

are

not

_{simultaneously performed.}

The Golden et al

_algorithm

_requires

more _memory,

because the list of

_offspring

needed and the

_tracing

of descendants

_requires

memory.

However,

the _memory

_required

is still linear with the number of animals N. Tier’s

₍₁₉₉₀₎

_algorithm

was faster than the

_{algorithm presented}

_here,

when

40 _years are

evaluated,

ie evaluation of 12.3

_generations

of animals. When _many

generations

are

evaluated,

_many animals have common _{ancestors, whose}

pedigree

is re-evaluated many times in the

present

algorithm.

Tier’s

_algorithm

avoids redundant calculation

by tracing pedigrees

once. The _memory

required

is 15.9 lVlb

for 28171 animals. The

_{presented algorithm required}

0.7 NIb is this situation. When about 6 or fewer

_generations

are

evaluated,

the

_present

algorithm

is faster

than

_Tier’s,

because Tier’s

_algorithm

first determines which elements of A are

required

before

_{calculating inbreeding}

coefficients. If

_only

the animals born in year

40 are

_evaluated,

the

_algorithm

_presented

here and that of Tier

₍₁₉₉₀₎

have about the same

_speed

(table II).

In Tier’s

_algorithm,

the

_required

elements of A have to be determined for

_relatively

few animals. The time

_{required by}

the

algorithm

of Golden

et al

₍₁₉₉₁₎

_equals

that

_required

if all animals are

_evaluated,

because the

_algorithm

re-calculates all

inbreeding

coefficients. The last alternative in table II shows that the presence of

large

half sib families is

advantageous

to Tier’s

_algorithm.

This is

because Tier’s

_algorithm

traces the

_pedigree

of the sire

_only

once.

DISCUSSION AND CONCLUSIONS

The

presented

algorithm

for

_computing

_diagonal

elements of A is a _sparse

imple-mentation of an

algorithm presented by

Mrode and

Thompson

(1989)

and

Quaas

(1989)

for the

_{multiplication}

of A with a matrix.

Here,

this matrix is the

identity

matrix and

_{only diagonal}

elements are calculated.

The

_algorithm

combines

_high

_{computational speed}

with low _memory

require-ment,

if the number of

_generations

evaluated is not

_larger

_than,

_say 12

_{(table II).}

In order to

_keep

the calculations within the

_physical

_memory of the computer,

none of the

_populations

evaluated in table II were

_{really large.}

If the data set

were much

larger,

Tier’s

(1990)

algorithm especially

would

_require

disk

memory

which would decrease its

_speed

_{considerably. However,}

the presence of

large

half sib

families favours the use of Tier’s

algorithm

(table II).

If the number of

_generations

involved

exceeds,

say

12,

the

_{algorithm presented}

here becomes slow

compared

to

Tier’’s,

because common ancestors are traced _{many times.} In order to overcome

this

_problem

we _may

_{compute F}

_i

=

1/2AS!d! _

L

L!,kLd:kDkk

and store the

k

sire’s row of L

(L

Sik

,

k =

1, ... ,

).

s

i

However,

calculating

the dam’s row of L costs

approximately

the same amount of

computer

time as

calculating

the individual’s

row of

L,

as in

!1].

] .

In

_practice,

often

_inbreeding

coefficients of _many animals in the

_population

have been calculated.

_Only

those of a few new born animals are unknown. The

_presented

APPENDIX

Integer

arrays

(N

is the number

of

animals in the

population):

PED(1:2, l:l!T)

PED(1, i)

= sire identification _no of animal i

PED(2, i)

= dam identification no of animal i

POINT(1:N)

POINT

_(i)=

the next oldest ancestor in the link list

= 0 if i is the last ancestor

Real

_Arrays:

F(0: N)

F(i)

=

inbreeding

coefficients of animal i.

F(O) =

-1

gives

appropriate

within

family

variances for unknown

parents

by

the formula:

_{.5-.25*(F (PED(1, i)+PED(2,i)).}

_F(i)

is

_initially

set

_equal

to

_-1,

which is more accurate than

calculating

F + 1 and then

_subtracting

1.

L(1:N)

L(j)

= element

_ij

of matrix

_L,

if animal i is evaluated

D(1:N)

D(i)

= within

_family

variance of animal i

ACKNOWLEDGMENTS

We are indebted to

Thompson

and referees for

helpful suggestions

and comments. The IB!Iilk

_Marketing

Board of

_England

and

_Wales,

the Meat and Livestock Commission and the

_Ministry

of

_Agriculture,

Fisheries and Food are

_{gratefully acknowledged}

for their

sponsorship.

REFERENCES

Golden

_BL,

Brinks

_JS,

Bourdon RM

₍₁₉₉₁₎

A

_{performance programmed}

method for

_computing

_inbreeding

coefficients from

_large

data sets for use in mixed-model

analyses.

J Anim

_Sci,

_69,

3564-3573

Henderson CR

_(197G)

A

_simple

method for

_computing

the inverse of a numerator

relationship

matrix used in the

prediction

of

_breeding

values. Biometrics

_32,

69-83

IMSL

₍₁₉₈₄₎

International Mathematical and Statistical Libraries.

_{Houston, TX,}

ed 9.2

Meuwissen

_THE,

van der Werf JHJ

₍₁₉₉₁₎

Effects of

heterogeneous

within-herd

variances on the

efficiency

or

dairy

cattle

breeding

schemes.

42th

Ann Meet Eur

Assoc Anim

_{Prod, Berlin,}

_Germany

Mrode

_R,

_Thompson

R

₍₁₉₈₉₎

An alternative

_algorithm

for

_{incorporating}

the

Quaas

RL

₍₁₉₇₆₎

_Computing

the

_diagonal

elements and inverse of a

large

numerator

relationship

matrix. Biometrics

_32,

949-953

Quaas

RL

₍₁₉₈₉₎

Transformed mixed model

_equations:

a recursive

algorithm

to eliminate

A-

1

.

J

_Dairy

Sci

_72,

1937-1941